A normal basis theorem for infinite Galois extensions

MATHEMATICS Proceedmgs A 88 (2), June 17, 1985 A normal basis theorem for infinite Galois extensions

by H.W. Lenstra, Jr.

Mathematisch Instituut, University of Amsterdam, Roetersstraal 15, 1018 WB Amsterdam, the Netherlands

Commumcated at the meeting of November 26, 1984

ABSTRACT

The normal basis theorem from Galois theory is generahzed to infinite Gaiois extensions 1NTRODUCT1ON

Let K be a field, L a Galois extension of K, and G the Galois group of L over K. We consider G äs a topological group with the topology defined by Krull [4; l, Chapitre V, Appendice II].

The normal basis theorem asserts that if L is finite over K there exists xeL such that the elements σ(χ), σ e O, form a basis for L äs a vector space over K, see [5; 3]. If L is infinite over K then no such basis exists, since for every xeL the sei {o(x):aeG} is finite. Hence if we wish to generalize the normal basis theorem to infinite Gaiois extensions we must look for an alternative formulation.

Let L be finite over K, and write (G, K) for the ÄT-vector space of all functions f:G^K. We let G operate on (G, K) by (σ/)(τ) =/(σ~ 'r), for σ, τ eG. We can now reformulate the normal basis theorem by saying that there is an iso-morphism q>:(G,K)~>L of A'-vector spaces that respects the action of G. Namely, if (a(x))akG is a basis of L over K, then we can define φ by φ(/) =

= Σσ εο /(σ)σ(*)· Conversely, if <p:(G,K)^L is an isomorphism äs above, and h:G->Kis defined by Λ(1) = 1, Λ(τ) = 0 (reG, τΦΐ), then χ = φ(Η) has the property that (σ(χ))σεΟ is a basis of L over K.

Galois extensions äs well, provided that we only consider continuous functions

THEOREM 1. Lei KcL be a Galois extension of fields, with group G, and denote by C(G, K) the K-vector space of all continuous functions f : G~*K; here G is provided with the Krull topology and K with the discrete topology. Lei G operate on C(G,K) by (σ/)(τ)=/(σ~'τ), for σ, τ eG. Then there exists an

isomorphism C(G,K)->L of K-vector spaces that respects the action of G.

The proof of this theorem is given in Section 3 of this paper.

We can also express the normal basis theorem by saying that, for L finite over

K, the additive group of L is free on one generator äs a left module over the

group ring K[G]. This assertion can be generalized to the infinite case äs follows.

Denote by U the set of open normal subgroups of G. We order t/by letting 7V' < 7V if and only if NC N'. For 7V, 7V' e U, 7VC 7V', let the ring homomorphism

ρΝ,/Ν:Κ[Ο/Ν]-*Κ[Ο/Ν'] be induced by the natural group homomorphism

G/N-+G/N'. We write K[[G]] = limNeU K[G/N], the projective limit being

taken with respect to the maps ρΝ>/Ν (see Section l for our conventions about projective limits). Observe that /f[[<j]] contains the group ring K[G] in a natural way, and is equal to it if G is finite.

For TVe U, let the subfield LN of L be defined by LN={yeL:a(y)=y for all σ ε 7V}; this is a finite Galois extension of K with group G/N. For 7V, 7V' e U, 7VC7V', the trace map TrN7N:LN-+LN> is defined by TrN7N(y) = ΣσεΝνΝ σ(^)· The projective limit UmNeU LN, taken with respect to the maps ΎτΝ7Ν, is in a natural way a left module over Äj[G]].

THEOREM 2. For any Galois extension of fields KcL with group G, the left K[[G]]-module limWe(y LN is free on one generator.

The proof of this theorem is given in Section 3 of this paper. 1. PROJECTIVE LIMITS

A preordered set is a set / with a binary relation < on / that is transitive and reflexive. A directed set is a preordered set / with the property that for any two a, ßel there exists yel with a<y and ß<y. A projective system consists of a directed set /, a set Ea for each cue/, and a mapfaß:Eß->Ea for each pair a, ßel with a<ß, such that faa equals the identity on Ea for each ael, and faßfßy=fay for all ct,ß,yel with a<ß and ß<y. The projective limit of such

a system, denoted by lim Ea or l i ma e / Ea, is defined by

lim Ett = { (xa)a 6 , e Π Ea : faß(Xß) = xa for all a, ßel with α < β} . αεί

The projective limit may be empty, even if all Ea are non-empty and allfaß are

Let 7, (Ea)ael, (faß) a βςΐ,α^β be a projective System in which all Ea are

non-empty. We suppose that for each ael we are given a collection (Ja of subsets of Ea, such that the followmg four conditions are satisfied.

(1.1) If a e / a n d >C r/tt then ^}Me

In particular, takmg / = 0, we see that Eae f/a.

(1.2) If a e / , and /C !/„ is such that f]Mey ΜΦ0 for all fimte subsets / ' C Λ then {~}Me

(1.3) If a, /Je/, a</? and x e £a, then/^'xe /^.

(1.4) If a, ßel, a<ß and Me (/ß, then /ö/ß[M] e f/a.

In the followmg proposition we wnte E=hm Ea. For a e / , we denote the

natural map E-+Ea by /„, and we put E'a= Π/?6/,«</? /«/?[£>]; so E'acEa, and

E'a~Ea if all/a;3 are surjective.

(1.5) PROPOSITION With the above hypotheses and notation, we have.

(a)

(b)/0[JS]=£;/0reecA«e/;

(c) ifJd is directed wiih respect to the restnction of < ίο /, then the image

of the natural map h ma e / Ea-^limasJ Ea is Iima e 7 E'a.

PROOF We need a few facts from the proof of [2, III. 7 4, Theoreme 1]. Let Σ denote the set of all famihes (Aa)aej for which

AaJ=0 and Aae 9a, for all ael, faß[Aß]cAa for all a, ßel, a<ß.

Let (Aa)ae,<(Aa)aeI if and only if A'aCAa for each ael. This makes £ into

a partially ordered set. In [2, /oc c;/ ] it is shown that Σ satisfies the conditions

of Zorn's lemma, and that the map E-> Σ sendmg (xa)ae/ to ({xa})ael estabhshes a bijection between E and the set of maximal elements of Σ

-We use this to prove (c). Let Jd be directed. It is trivial that the image of

E m hmaeJEa is contamed in hmaej E'a. To prove the other mclusion, let (Xa)aejehmasJE'a.ForßeI, letA(,= C]aej,a^ßfaß}xa. We claim that ^ ^ 0 .

To prove this, it suffices by (l 3) and (1.2) to show that f]aeK,a<ß /αβίχα*® for every fimte subset KcJ. Let K be such We may assume that ΚΦ& Smce

J is directed, we can choose yeJ such that a s y for all aeK, and smce I is

directed we can choose Sei such that β<δ, γ<δ. We have xyeEyCfyo[Eä],

so x,,=/),^(z) for some zeE?, and it is now readily venfied that fßg(z)e

e Γΐαεκ,α^β faßlXa· This proves that ΑβΦΰ. It follows that (Ap)ßele Σ The results about Σ quoted above imply that Σ h as a maximal element ({yß})ßel Wlth ({^J^e/ai/lysVe/· T n e n ^/5e^/9. a n d s m c e ^«={^«1 f°r α ^ / thlS

imphes that ^α=Λ:β for all ae J. Hence (yß)ßSieE maps to ( χ ^ ^ , / ε ΐ ι π ι , ^ £α.

This proves (c).

PROOF With Zorn's lemma, choose WC V maximal among all subsets

W'cV for which the sum N+ ®„eW Mw is direct. Then for we V the sum

(N+@weWMw) + MO is not direct, by the maximality of W, so

(N+®weWMw)r\Mu*Q; but Mu is simple, so (N+®WEWMw)nMu = M„,

and M0CN+ ®weWMw This imphes that M = N+ ®weWM„, and the lemma follows easily This proves (2 3).

(2.4) LEMMA Lei R be an Artin ring, and xeR Then \ve have:

(a) χ is a unit if and only ifii is a left umt, and if and only if it is a nght umt; (b) R has only fimtely many maximal two-sided ideals.

(c) χ e R * if and only if (x mod m) e (R/m) *for every maximal two-sided ideal m of R.

PROOF (a) It suffices to show that yz=l imphes zy=l. The descendmg cham condition imphes that Ry" = Ry"+' for some n>0, so y" = wy"+i for some weR Then l =y"z"= wy" + lz" = wy and w = wyz = z.

(b) Let mι,τη2, ··.,!% be distmct maximal two-sided ideals. Then

m, + mj = R for i^j, so the map Α->Π,*_ι ·#/«!, is surjective with kernel

Πί_ι m,· This proves that P|f i m; 1S properly contamed m p)fj,' m,. The

descendmg cham condition now imphes a bound on k.

(c) "Onlyif" is clear. To prove "if", suppose that χ $ R *. ThenRx^Rby (a), so RxcL for some maximal left ideal LcR. Let m = Ann (R/L)CR be the annihilator of the simple Λ-module R/L. Then mCL so (R/m)(x mod m)C

CL/m^R/m and consequently (x mod m)$(jR/m)*. Hence to prove (c) it

suffices to show that m is maximal äs a two-sided ideal. We have m = C}yeR_LLy where Ly = {reR: ryeL}; considermg the map R^R/L sendmg l to y one finds that R/Ly~R/L äs /?-modules. By the descendmg cham condition we have m= ClyeT Ly f°r somefmite set TcR- L. Then R/m

is a submodule of RyeT R/Lys(R/L)*T, so R/m = (R/L)m for some m>0, by (2.3) Let now n be a two-sided ideal of R contaming m. Then R/n = (R/L)" for some« > 0 , by (2 3). If « = 0 then n = R, and if n > 0 then n = Ann (R/n) =

= Ann (R/L) = m This proves (2 4).

(2.5) LEMMA Lei g'R0-*R] be a surjective ring homomorphism from an

Artin ring R0 to a ring RI, and let aCR0 be a two-sided ideal. Then

PROOF The mclusion C is obvious. To prove D we first suppose that aCm for every maximal two-sided ideal m of R0 that does not contam ker g. Let x= l +#00 ε (l +5Γ[ο])Π/?*> w u h y e a . Usmg (2 4)(c) we prove that l +yeR^.

Let m be a maximal two-sided ideal of RQ If ker g Cm then the natuial map

R0^-R0/m factors via g, so x = g(\ +y)eR* imphes that (l +y mod m)e

e(RQ/m)*. If ker g(£m then j e a C m by hypothesis, so (l +y mod tn) =

= (1 mod m)e(/?0/m)*. This proves that 1+yeRQ, so x = g(l+y)c

Assertions (b) and (a), which form [2, III 7 4, Theoreme 1], follow from (c) by puttmg / = { « } and J=0, respectively This proves (l 5)

2 ARTIN RINGS

Rings are supposed to have unit elements, and ring homomorphisms are supposed to preserve these The group of umts of a ring R is denoted by R * A projecüve System of rings is a projective System /, (Ra), (gap) m which each

Ra carnes the structure of a ring and each ga/1 is a ring homomorphism The projective hmit of such a System carnes a natural ring structure

The followmg proposition is not needed in the sequel, but its proof motivates the approach taken later

(2 1) PROPOSITION Let I, (Ra), (gaß) be a projective System of rings in

which each Ra satisfies the descendmg chain condition on two-sided ideals and

each gaß is surjective Put R = \im Ra Then the natural map R~*Ra is

sur-jective for every ael

PROOr We apply (l 5) with Ea = Ra, faß = gaß and

f/a = {0}l){x+a. xeRa, aCRa is a two-sided ideal}

It is clear that Εαφβ and that (l 1), (l 3) and (l 4) are satisfied To prove (l 2), we note that the descendmg cham condition on two-sided ideals of Ra imphes the existence of a minimal element among all fmite mtersections of sets M e / , this minimal element must then be Π/v/e / - ^

Since the gajl are surjective we have E'a = Ea = Ra m (l 5), so (2 1) follows from (l 5)(b) This proves (2 1)

An Artin ring is a ring that satisfies the descendmg cham condition on left ideals

(2 2) PROPOSITION Let I, (Ra), (gaft) be a projective System of rings m

which each Ra is an Art m ring and each gaß is surjective Put R = hm Ra Then

the natural map R*~+R* is surjective for each ael

The properties of Artm rings needed m the proof are listed m Lemma (2 4) This lemma can easily be denved from the structure of semisimple Artm rings and properties of the Jacobson radical We give a direct proof, starting from the followmg well-known lemma By a module we mean a left module on which the umt element acts äs the identity, and a module is called simple if it is non-zero and has no submodules except itself and {0}

(2 3) LEMMA Let R be a ring, (Mv)VEy a collection of simple R-modules, M- ®uevMu, and NCMa submodule Then there is a subset Wc V such that

In the general case, let τηΐ5ηΐ2, ...,mk be the maximal two-sided ideals of R0 that do not contain ker g. Then m, + ker g = Ro for each /, so some z, em, satisfies z,^l mod ker g. Then z = z\Z2---zk satisfies zem\m2---mk and

g(z) = l. Therefore #[α]=£[&] for 6 = amim2---m/(. Applying the previous case

to b we see that (l +g[a])r\Rf = g[(l +b)nÄ0*]Cg[(l +a)n/?0*]· This proves (2.5).

(2.6) LEMMA. Let g:R0-+R\ be a surjective ring homomorphism from an

Artin ring R0 to a ring /?,. Then the map RQ-+R* induced by g is surjective. PROOF. Put a=R0 in (2.5). This proves (2.6).

PROOF OF (2.2). We apply (1.5) with Ea = R* and faß-.Rß-*R£ the map

induced by gaß. For ya we take

ya = {0}ö{(x+a)CiR*: x<=R*, aCRa is a two-sided ideal}.

It is clear that ΕαΦ0. We check conditions (1. !)-(!. 4).

(1.1) If C\M€y-M^0 then with xef}MeyM each M e "7 has the form

(x + aM)C\R* for some two-sided ideal aMCRa, and then (~}Me?-M=

= (x+a)r\R* where α = Γ\Με?<ϊΜ·

(1.2) Using (1.1) we may assume that i^is closed under taking finite inter-sections. For M e "7, let bM be the two-sided ideal of Ra generated by

[y-Z'.y, zeM}; then M=(x + hM)r\R* for each xeM. Choose M' 'e "7 such that bW' is minimal among all ideals bM, M&:f. It then follows that

M'^{\M,,M. (1.3) This is clear.

(1.4) If Μ=(ΛΓ+ο)Π/?/ε cfß then by (2.5) we have

faß[M] =faß[x((l + ο)Π/?/)] =faß(x) · ((l +6,/,[α])ΠΑ *) =

From (2.6) we see that the m a p s / ^ are surjective, so E'a = R* in (1.5). The proposition now follows from (1.5)(b).

This proves (2.2).

(2.7) PROPOSITION. Let I, (Ra), (gaß) be a projective System of rings in

which each Ra is an Artin ring and each gaß is surjective. Let fürt her I, (Ma),

(haß) be a projective System in which each Ma is a free Ra-module on one

generator and each haß is a surjective Rß-module homomorphism; here Ma is

considered äs an Rß-module via the map Rp-^Ra. Put jR=lim Ra and

M=lim Ma. Then M is a free R-module on one generator, which can be

chosen of the form (xa)ae!, where each xa generates Ma äs an Ra-module.

PROOF. Without loss of generality we may assume that Ma = Ra, for each

. Letfaß:R^R* be the map induced by haß\ then the Statement of (2.7) is equivalent to the assertion that lim R*=£0, the projective limit being taken with respect to the maps faß. *~

To prove that lim R*^0 we apply (1.5) with the same Ea and ,</a äs in the proof of (2.2); butfaß differs from the mapfaß used for (2.2) by a unit factor

haß(l) on the right. Since .7α is transformed into itself by multiplication by units conditions (!.!)-( 1.4) are still satisfied. The proposition now follows from (1.5)(a).

This proves (2.7). 3. GALOIS EXTENSIONS

In this section we use the notation from the Introduction.

PROOF OF THEOREM 2. We apply (2.7) to the projective system U, (K[G/N]),

(ΘΝ'/N) of rings and the projective System U, (LN), (ΎτΝ7Ν) of modules. Each

K[G/N] is finite dimensional over K and therefore an Artin ring. Each LN is free over K[G/N] on one generator, by the normal basis theorem. The re-maining conditions are easy to check. Theorem 2 now follows from (2.7). PROOF OF THEOREM 1. From (2.7) we obtain an element (xN)NeUe such that

(3.1) (a(xN))aeG/N >s a basis of LN over K, for each Ne Ut (3.2) ΊτΝ,/Ν(χΝ)=χΝ· for N, N'e U, NcN'.

To define <f>:C(G,K)->L, letfeC(G,K). Since K is discrete, there is for every σ ε G an Ne U such that / is constant on σΝ. By compactness of G, we can choose the same N for all σ. Let fN:G/N-+K be the map induced by /. We now put

<?(/)= Σ /Ν(σ)σ(χΝ).

aeG/N

From (3.2) it easily follows that the express'on on the right does not depend on the choice of N, so φ is well-defined. It is also ^-linear, and it respects the action of G. Finally, (3.1) implies that it is bijective.

This proves Theorem 1.

Let L be finite over K, and let M be an intermediate field that is also Galois over K. Applying the trace from Lto M one obtains, from every normal basis of L over K, a normal basis of M over K. In addition, every normal basis of

M over K can be obtained in this way, since the natural map K[G] *-*K[G/N] *

is surjective (Lemma (2.6)); here M=LN .

The extension of these results to the general case is äs follows. Let L again

C(G,K)~+L äs m Theorem l one obtams, upon takmg invanants undei N, an isomorphism C(G/N,K)^M such that the diagram

C(G/N,K)

C(G,K) > L

is commutative; here the first vertical arrow is mduced by the canonical map G-^-G/N, and M-+L is the mclusion. Conversely, given an isomorphism C(G/N,K)-+Mof K-vecioi spaces that respects the action of G/N one can find an isomorphism C(G,K)—>L äs m Theorem l such that the above diagram commutes This is a consequence of (l .5)(c), with / = U and J equal to the set of open normal subgroups of G that contain 7V.

REFERENCES

1 Bourbaki, N - Algebre, Chapitres 4 et 5, Hermann, Paris, 1959 2 Bourbaki, N - Theorie des ensembles, Hermann, Paris, 1970

3 Deunng, M - Galoissche Theorie und Darstellungstheone, Math Ann 107, 140-144 (1933) 4 Krull, W — Galoissche Theorie der unendlichen algebraischen Erweiterungen, Math Ann

100, 687-698 (1928)

5 Noether, E - Normalbasis bei Korpern ohne höhere Verzweigung, J reine angew Math 167, 147-152 (1932)

Journal of Pure and Applied Algebra 36 (1985) 281-298 281 North Holland

ABELIAN VARIETIES HAVING PURELY ADDITIVE REDUCTION

H.W LENSTRA, Jr.

Mathematisch Instituut, Roeteisstraat 15, 1018 WB Amsterdam, The Netherlands F. OORT

Mathematisch Instituut, Budapestlaan 6, 3584 CD Utiecht, The Netherlands

Communicated by F Oort Received 30 November 1983 Revised II April 1984

Let E be an elhptic curve over a field K with a discrete valuation v with residue class field k. Suppose E has 'additive reducüon' at v, i.e. the connected component AQ of the special fibre A0 of the Neron minimal model is isomorphic to Oa. Then the order of A0(k)/Ao(k) is at most 4 äs can be seen by inspection of the usual

tables, cf. [9, pp. 124-125] and [5, p. 46]. Thus it follows that if the order of the torsion subgroup Tors(E(K)) is at least 5 and pnme to p = char(^), the reduction cannot be additive. This note arose from an attempt to see whether an explicit classificaüon really is necessary to achieve this result This attempt turned out to be successful: we prove a generaiization for abelian vaneties (cf. 1.15). The proof does not use any specific classjfication, but it relies on monodromy arguments. It explams the special role of pnme numbers / with l<2g+ l in relation with abelian vaneties of dimension g. Note that Serre and Täte already pomted out the importance of such pnmes, cf. [14, p. 498, Remark 2]. In their case, and in the Situation considered m this paper the representation of the Galois group on T/A has dimension 2g, hence pnmes / with t<2g+ l play a special role

We give the theorem and its proof m Section 1. Further we show that the bound in the theorem in sharp (Section 2), and we give examples in Section 3 which show that the restnction /=£chai(/c) m the theorem is necessary. In Section 4 we indicate what can happen under the reduction map E(K)->E0(k) with pomts of order p in

case of additive reduction.

K. Ribet made several valuable suggestions on an earher draft of this paper The elegant methods of proof in Section 2 were suggested by him. We thank him heartily for his interest m our work and for his stimulatmg remarks

282 H.W. Lenstra, Jr., F dort

1. Torsion points on an abelian variety having purely additive reduction

Let K be a field and υ a discrete valuation of K. We denote the residue class field of D by k; we assume k is perfect. Let Ks be a separable closure of K and 0 an extension of υ to Ks. We denote the inertia group and first ramification group of

ΰ by / and J, respectively. These are closed subgroups of the Galois group G&\(K^/K). If the residue characteristic char(Ä:)=p is positive, then / is a

pro-p-group; if char(£) = 0, then J is trivial. The group J is normal in /, and the group I/J is pro-cyclic:

///=

Π ζ/·

/pnmc, /7Hhar(/t)

Let A be an abelian variety of dimension g over K, and </ the Neron minimal model of A at v, cf. [9], We write A0 for the special fibre: A0= t/(x)Ä k, where 7? is the valuation ring of υ. We denote by AÜQ the connected component of A0. Let

be the 'Chevalley decomposition' of the £-group variety A%, i.e., B is an abelian variety, Ls is a torus, and La is a unipotent linear group. We write

We say that A has purely additive reduction at υ ii LU=AQ, so if α=μ = 0 (and we

say additive reduction if dim/1 = l = dimLu).

Throughout this paper, / will stand for a prime number different from char(£). If G is a commutative group scheme over K, and n e Z, we write G[n] for the group scheme Ker(«· 1C:G-*G), and

This is a module over the ring Z, of /-adic integers, and it has a continuous action of Ga\(K,/K). For G = Gm, the multiplicative group, T,G is free of rank l over Z,,

and the subgroup !CGa\(Ks/K) acts trivially on T,Gm. We write

U,= T,A.

This is a free module of rank 2g over Z/.

Let M be a finitely generated Z/-module. By the eigenvalues of an endomor-phism of M we mean the eigenvalues of the induced endomorendomor-phism of the vector space M®Z/(Q/ over the field Q/ of /-adic numbers. Suppose now that M has a

continuous action of 7. If 7'C/ is a subgroup, we write M7'={A:6M:T^ = ^for all r e / ' } .

We claim that the image /0 of J in Aut(M ) is finite. If char(/t) = 0 this is trivial, so

Abehan vaneties having purely additive reduction 283

has trivial mtersection with this kernel, so JQ is isomorphic to a subgroup of Aut(M//M) and therefore fmite This proves our claim

We defme, m the above Situation, the averagmg map Nj M-+MJ by

This map is the identity on MJ, so gives nse to a Splitting

M=MJ@kerNj (l 1)

It follows that the functor ( )J is exact

(l 2) Notice that MJ has a contmuous action of the pro cychc group /// This is m par-ticular the case for

We denote by σ a topological generator of I/J

1.3. Proposition. The multiplicity of l äs an eigenvalue of the action of σ on

X/ = Uf is equal to 2μ + 2a In particular, it does not depend on the choice of the pnme number l Φ char(/0

Proof. We begm by recalhng the results from [SGA, 7 I( exp IX] that we need, see

also [11] Let a polanzation of A over k be fixed Then we obtam a skew-symmetnc painng

which is separattng in the sense that the induced map t// -»Hörn/, (t//, 7}(B,n)

becomes an isomorphism whea tensored with Q/ The painng is Galois-mvaiiant in the sense that

(TM, TL>> = r<«, D) for τ e Gal(K^/K), u,ueUh

= (u,u} ifre/

We wnte

v=u,', w=vr\v^,

where -L denotes the orthogonal complement m U/ with respect to < , ) We have

rank///W=ß, rank/lV/W=2a (14)

Smce A has potentially stable teduction, there is an open normal subgroup 7'C/ such that the module V- UJ satisfies

284 H. W. Lensira, Jr., F. Oort

We now take ./-invariants. The Galois-invariance of < ·, · > implies that X/ = Uf is orthogonal to the complement of U/ in t// defined in (1.1). Therefore <·, · > gives rise to a separating Galois-invariant pairing

which will again be denoted by <·,·>. We let § denote the orthogonal complement in X, with respect to <·,·>.

There is a diagram of inclusions

where μ and 2a indicate the Zrranks of the quotients of two successive modules in

the diagram; here we use (1.4) and the equalities

which follow by duality.

All eigenvalues of σ on Kare l, and by duality the same is true for X//V\ hence for X,/W*. We have

rank/( V

so in order to prove the proposition it suffices to show that

no eigenvalue of σ on W^/V equals 1. (1.6)

Let 7 = V'J. We first prove that

no eigenvalue of σ on Y/V equals 1. (1.7) Suppose in fact, that ye Fsatisfies ay=y + v for some ue V. Then a"y=y + m for all positive integers «. Choosing Λ such that σ " ε / ' we also have o"y-y, since

ye V, so we find that i> = 0 and ye V. This proves (1.7).

We have 7§C 7, by (1.5), so (1.7) implies that

no eigenvalue of σ on (Y§+ V)/V equals 1. (1.8)

By duality, (1.7) implies that no eigenvalue of σ on F§/ 7§ equals l, and therefore

no eigenvalue of σ on (V^ + K)/(7§+ K) equals 1. (1.9)

From W- K D F§ it follows that V^+Vis of finite index in W^, so (1.8) and (1.9)

Abelian vaneties having purely additive reduction 285

1.10. Corollary. The abelian variety A has purely additive reduction at υ ij 'and only

if σ has no eigenvalue equal to l on X.

Proof. Clear from Proposition 1.3. It is easy to prove the corollary directly, using

that rank Υ= + 2α. Π

Let 7 ' c / and Y=(U'')JCXi be äs in the proof of Proposition 1.3, and n a positive integer for which σ" e Γ . Then σ" acts äs the identity on Y, and by duality also on X//Y§. By F§C Υ this implies that all eigenvalues of σ" on X/ are 1. Thus we find that all eigenvalues of σ on X/ are roots ofunity. These roots of unity are of order not divisible by char(A:) =/?, since the pro-/?-part of the group I/J is trivial. Let a/(m) denote the number of eigenvalues of σ on X/ that are m-th roots of unity, counted with multiplicities.

1.11. Proposition. For any two prime numbers l, l' different from char(£) and any

positive integer m we have a/(m) = a/>(m).

Proof. We may assume that m is not divisible by char(^). Let L be a totally and

tamely ramified extension of K of degree m. Replacing K by L has no effect on J, but σ should be replaced by σ"1. Since a/(m) is the multiplicity of l äs an eigenvalue of σ'" on X,, the proposition now follows by applying Proposition 1.3 with base

field L. D

1.12. Corollary. The number rank^A'/ does not depend on l. Proof. This follows from Proposition 1.11, since

rankz/ X, = sup,„ a, (m). D

Remark. Proposition 1.11 and Corollary 1.12 can also easily be deduced from the

fact that, for each rel, the coefficients of the characteristic polynomial of the ac-tion of r on U/ are raac-tional integers independent of /, see [SGA, l I, exp. IX, Theoreme 4.3].

1.13. Theorem. Suppose that A has purely additive reduction at υ. Then for every

prime number l φ char(k) the number o(/)e {0, 1,2, ...,00} defined by

is finite, and

286 // W Lmstra Jr F Oort

Proof. First let / be a fixed pnme, /^char(Ar), and iet N be a positive integer We

have

= #(kernel of σ - l on / I f / " ] ^ ) ' ) = #(cokernel of σ - l on A[1N}(K,)'),

the last equahty because A[1N](K^ is fmite By (l 2) the natural map

is surjeclive, so the above number is < #(cokemel of σ - l on X()

Let us wnte |/ for the normahzed absolute value on an algebraic closure Q/ of |/ =

#(cokernel of σ - l on X,)= jdet(a~ l on X{)\,}

for which |/|/ = / ! Then by a well-known and easily proved formula we have

!

where ζ ranges over the eigenvalues of σ on Xt

Lettmg N tend to mfmity we see that we have proved

By Corollary l 10 the nght hand side of (l 14) is finite This proves the claim that

b(l) is finite

Next we exploit the fact that the eigenvalues ζ of σ are roots of unity It is well known that for a root of unity ζφ\ we have

\ζ-\\,>1 υ(ί " if ζ has / power Order, | ζ --1|,= l otherwise

Wnte o/(/°°) = max;Vi7/(/'v) Then (l 14) imphes that

so there is a number d(l) such that

(/-Now let q be an arbitrary pnme number different from char(/c) Usmg Proposi-tion l 1 1 we deduce Σ (/-1)6(/)^Σ«/(^( / )) /pnme. H Idr(i-) /

= Σ °

(i

)

Abelian vaneties having purely additive leduction 287

1.15. Corollary. Suppose that A has purely additive reduction at v. Denote by m

the number of geometric components ofthe special fibre A0 of (he Neron minimal

model of A at υ. Then

X (/-l)ord,(/n)<2£

/ prime, /=£{.har(£)

where ord/(m) denotes the number of factors l in m.

Proof. Analogous to the proof of [11, 2.6]. Π

We shall see in Section 3 that the restriction /^char(/c) is essential in Theorem 1.13. We do not know whether this is also the case for Corollary 1.15.

1.16. Remark. In [17] we find a weaker version of the result mentioned in Corollary

1.15.

2. An example which shows the bound in Theorem 1.13 to be sharp

2.1. Example. Let / be an odd prime number, and g = (l- l)/2. We construct an

abelian variety A of dimension g over a field K with a point of Order / rational over

K such that A has purely additive reduction at a given place of K.

Let ζ = ζ, be a primitive /-th root of unity (in C), and F : = Q(£). We write £> = Z[£] for the ring of integers of F. The field F0: = Q(C+C) is totally real of

degree g over Q and F is a totally imaginary quadratic extension of F0, i.e. T7 is a

CM field. We choose

in this way, cf. [15, 6.2 and 8.4(1)], we obtain an abelian variety

with End(B) = D, with a polarization λ:Β->Β' (defined by a Riemann form, cf. [15, p. 48]):

AutCB,A)=<Ox{+l}=//2/;

in fact by a theorem of Matsusaka, cf. [3, VII.2, Proposition 8], we know that Aut(5, A) is a finite group, hence only the torsion elements of the group of units of Z[£] can be automorphisms of (B, λ), moreover complex multiplication by ζ leaves the Riemann form invariant (use [15, p. 48, line 8]), and the result follows. LetPeß

be the point

P =

288 H W Lenstra, Jr, F Oorl

note that l —ζ divides /eZ[£], hence P is an /-torsion point; moreover

i-f i-c

hence complex multiplication by ζ leaves P invariant; thus

By [15, p. 109, Proposition 26], wecan choose a number field K such that 5 is defin-ed over K, such that PeB(K), and such that AutK(B,P) = Z/l. We choose a prime number p such that

jD=l(mod/), and p\ discriminant(ÄYQ)

(by Dirichlet's theorem there exist infinitely many prime numbers satisfying the first condition). Let υ be a place of K dividing p. If B has bad reduction at υ we choose

A=B; if B has good reduction at υ we proceed äs follows. We have

thus there exists a (unique) field L with KcLcK^p), and Gal(L/X) We choose an isomorphism

α : Gal(L//0 ^ Aut^(ß, P) = H=Z/l.

By [12, p. 121] we know

H\G = Ga\(L/K), H=Aut

(B, P)) = Hom(G, H),

thus by [13, p. III-6, Proposition 5] this element α corresponds to a pair (A,Q)

defined over K such that

(A,Q)®KLsi(B,P)<S)KL.

We note that A has bad reduction at u: the extension L D A" is totally ramified at

ü, we assumed that B has good reduction at v, hence the inertia group / a t υ operates

tiivially on TpB, and by twisting with (the trivial) α we see that 7 operates non-trivially on TPA. Note that A ®KL has CM, thus A has potentially good reduction at all places of K. From these facts we deduce that A (in both cases considered) has purely additive reduction at υ äs follows; let A°0 be the connected component of the special fibre of the Neron minimal model of Λ at D; then

980

Abelian varieties havmg purely additive reduction

2.2. Remark. One can also construct an example with residue-characteristic zero. Consider (B, P) äs constructed above (say over k=C), choose a deformation of this over k[[T]] on which H =2/1 acts; then we obtain an abelian variety A defined over K=k((T))H, and P<=A(K) of order /; it is not difficult to see it has bad reduction

(at Γ'ι-·0). We leave the details to the reader.

2.3. Remark. We make (2.1) more explicit. Let / be an odd prime, / = 2g+ l, let p

be an odd prime, ρΦΐ, let Ä"=(Q(C/) and suppose a curve C is given by the two

af-fine curves deaf-fined by the equations

which are identified along the open sets (χφο) and (ξ*0) by

Χ=\/ξ, Υ=η/ξ8η.

Thus we have a complete (hyperelliptic) algebraic curve of genus g and

Χ~ζΧ, Υ -Υ, ζ = ζ, ξ "ξ/ζ, η~ζ,Άη

is an automorphism φ of order /. The points

define

P:=Cl(a-/J)e/l:=Jac(C).

We see that α and β are invariant under φ, thus />eJac(C) is invariant under 0*e Aut(,4). Note that Y-p defines a rational function on C; this function has /· α

äs set of zeros, its poles are not on the f irst affine curve, hence / · β is the set of poles;

thus la-lß~0, i.e. l · P = 0. The points of order 2 on^4 are generated by the points Cl(y-ß), where γ = (χ,0) and x1 +p2 = Q; thus we see that

operates non-trivially on poiuts of order 2 on A, and because this extension is ramified above each place υ dividing p, and because p φ 2, we conclude that A does not have good reduction ac v. Moreover

Z[(]CEndK(A)

and we conclude äs before. The last step can also be made explicit; choose a zero

*of X'+p2 = 0; then ye{('x i= l, . . . , / } , write Ql = Cl((C'x,0}-ß), and denote by

290 H W Lenstra Jr f< Oort

= (Z/2)2t is generated by Q,, , g, and the

ö i + + Qi = 0 Thus the action of ζ on Λ [2](j?) is given by

only relation is

Q/

has no eigenvalues equal to +1 and by Corollary l 10 (applied with the pnme 2) we conclude that A has purely additive reduction

3. Points of order p on eliiptic curves having additive reduction

Let K, u, and kbe äs m Section l, and supposechar(/c)=/?>0 Let A be an abelian

variety over K having additive reduction at D, we have seen in Theorem l 13 that the pnme-to-/? torsion m A(K) is very hmited m this case What about the p power torsion m this case9 With the help of some examples we show this torsion can be arbitranly large

First we give equal-charactenstic examples

3.1. Example. Let p^5 (modo) and suppose gi\en an integer / > ! We construct K, v, k, E such that char(Ä')=/? = char(yi:), E has additive reduction at υ and

p1 divides #(E[p'](K))

Consider /c = Fp and L = k(t), defme an elliptic curve C over L by the equation

27 a= — 4 1728-t ' note that y ( C ) = 1 7 2 8 - ^ ^ = i y i 4α3 + 27α2 '

and that its discnmmant equals

Abehan varieties havmg purely additive reduction 291 (its ./-invariant being integral), and it has bad reduction at w, because its discrimi-nant satisfies

note further that for any extension K DL of degree not divisible by 6 and for any extension υ of w to K the reduction at υ is additive (note that C is of type II = Cj at w, cf. [5, p. 46]). Let φ be the /-th iterate of the Frobenius homomorphism, and let M be its kernel:

thus E is given by the equation

Y2 = X3 + a"X+aq, q=p',

and M is a local group scheine of rank q. Note that C is not a super-singular elliptic curve (because its y'-invariant is not algebraic over k), thus

By duality we obtain

MD = NCE,

We take for K DL the smallest field of rationality for the points in 7V, and we extend w to a discrete valuation υ on K. Note that K D L is a Galois extension and the degree

[K: L] divides #(Aut(//<?)) = (/>- l)p'~l;

thus 3 does not divide [K: L], we conclude E(x)LK has additive reduction at υ; moreover

Z/p'cE(K)

by construction, and the Example 3.1 is established.

3.2. Example. Take p~=2, the other data äs in Example 3.1, and we construct E so

that

2' divides #(E[2'](K)).

Define C over L = k(t), /c=F2, by the equation

292 H W Lenstra, Jr , F Oort note that 3 does not divide

#(Aut(Z/2')) = 2'->, i>\,

and the methods of the previous example carry over.

Now we construct some examples in which char(A') = 0 < p = char(A:).

3.3. Example. Take p = 2, let i> l be an integer. We construct K, u, k, E äs before,

such that E has additive reduction at v, and such that char(K) = Q, char(A:) = 2, and E[2']CE(K).

Let m>\ be an integer, define

L = Q(n), 7 rm + l= 2 , νν(π)=1,

choose aeL, and let E be given over L by the equation

Y2 + nmXY=X3 + π2αΧ2 + αΧ; the point

Ρ = (-\/π2, l/n3)eE(L)

is a point of order 2, because it is on the line 2Y+ nmX = Q, and the same holds for (0,0) e ^ L ) ; thus E[2]CE(L).

Suppose w(a)> 1; because

we conclude w(A) = 4m + 2w(a); suppose

m=\ and w(a) = 2, thus w(A) = 8 and

or

m = 2 and w(ö) = l, thus w(/l)=10 and w(y)>0;

then the equation is minimal, the curve E has additive reduction at w and the reduc-tion is potentially good. Let K DL be the smallest field of rareduc-tionality for the points of E[2']\ note that

Gal(K/L) C Aut((Z/2')2) = GL(2, Z/2') is in the kernel of

GL(2,Z/2')->GL(2,Z/2)

(because £[2] CE(K) by construction), thus the degree [K : L] is a power of 2, hence it is not divisible by 3. This implies that υ(Δ) is not divisible by 12 (where υ is some

extension of w to K), thus the reduction of E®LK at u is additive (because of w(y)>0 it cannot become (Gm-type). Hence over Λ' we have

Abehan varieties having purely additive reduction 293

3.4. Example. Let p=5 (mod 6), and let / > l be an integer. We construct K, v, k, E

äs above with char(Ä') = 0<char(Ä:)=/', with E having additive reduction at υ, and

E[p']CE(K).

Consider over (Q the modular curve X0(p)<$', this is a coarse moduli scheme of pairs

NcE where E is an elliptic curve and N a subgroup scheme over a field K such

that N(K,) = Z/p; consider the scheme M0(p) over Spec(Z) (cf. [4, p. DeRa-94, Theoreme 1.6] and [6, p. 63]), and consider the point x0eM0(p)(fp) given byj = 0. Note that p = 2 (mod 3) implies that the curve E0 with y' = 0 is supersingular in characteristic p, hence it has a unique subgroup scheme ap = N0CE0, the kernel of Frobenius on E0. Let (} be the local ring of M0(p)®1 W at x0, where W= Wx(¥pi) (i.e. W is the unique unramified quadratic extension of Zp). We know: the local deformation space of ap = N0CE0 is isomorphic to the formal spectrum of

, Y]]/(XY-p),

the automorphism group Aut(E®fp^) = A' acts via

AV ±1=1/3

on W\\X, Y]]/(XY-p), and the completion of ^ is canonically isomorphic to the ring of invariants

ß=W[[S, T]]/(ST-pi), S = X\ T=Y3.

(cf. [6, p. 63] and [4, VI. 6]). Let L be the field of fractions of W (i.e. L is the unramified quadratic extension of <Qp), and construct

()-+Ö-*L by S »p2, T~p;

this is a point xeX0(p)(L); by results by Serre and Milne (cf. [4, p. DeRa-132,

Proposition 3.2]) we know there exists a pair NCE defmed over L. N®L&~1/p,

with moduli-point r. Let ΛΓ be the smallest field containing L such that all points

of E[p'] are ratioaal over K. Note that the degree [K: L] divides (/7-l)V, thus it is not divisibk by 3; hence

the pair (NcE)®K does not extend to a deformation of apCE0; it follows that E does not have good reduction at the discrete valuation υ of K (if so, N would extend flatly, reduce to a subgroup scheme of rank p of E0, hence to ap = N0cE0). Thus £ has additive reduction at υ, and by construction

294 H W Lenslra, Jr , l· Oorl

3.4 bis. Example. Consider p = 1 1 , take 121. H of [5, p 97] This is a curve E over L = Q with additive reducüon at w = un, with w(A) = 2, with w(y)>0 and which

has a subgroup scheine of order 11 Now proceed äs before K = L(E[\\']), etc., and we obtam a curve E over K with additive reduction at υ (a valuation lymg over

w), and with £'[11']C£1(Ä')

3.5. Remark. We have not been able to produce examples analogous to Example

3 4 m case p = l (mod 3). Hence for these primes the Situation is not clear; we did not get beyond an example of the following type·

3.6. Example. Take p = 7, consider a curve with conductor 49 over Q, cf [5, p 86]

Then w(A) = 3 or w(A) = 9 (with w—v-j), and the curve has potenlially good reduc-tion (because of CM); furthermore it has a subgroup scheme NcE over Q of rank 7 Thus K- = (Q(N) has degree dividing 6, we see that υ(Δ) is not divisible by 12

(where υ lies over w) thus E has additive reduction at υ and

3.7. Example. Consider p = 3, and let i>\ be an integer We constuct K, υ, k, E äs

before with char(/0 = 0, char(/c) = 3 and E[3']CE(K). We Start with L = Q, νν=υ3,

and we choose an elhptic curve E over Q with minimal equation / such that:

E has additive reduction dt w, w(j)>0, w(Af)=l (mod 2), and (Z/3)c£"(Q);

such examples exist, e.g see [5, p 87], the curve 54 A has w(A) = 3, w(y)>0, and Z/3 = E(Q) Let K = <$(E[3']\ then [AT:Q] divides 2-39, thus v(A)*Q (mod 4) for

any υ lymg over w = u3; thus

£" has additive reduction at u, and £'[3']C£1(Ä') 4. The image of a point of order p under the reduction map

Let A be an abehan vanety over a field K, let R c K be the ring defmed by a discrete valuation υ on K, and let </ be the Neron minimal model of A over Spec(/?).

At first suppose n > l is an integer such that char(Ar) does not divide n (here k is the residue class field of u, i.e & = /?/m). Let y[/i] denote the kernel of multiphcation by n on / Note that

y[«]-"Spec(/?)

is etale and quasi-fmite Thus we see that A(K)[n] mjects m A0(k) (here A0 = / ®f i A: is the special fibre), and all torsion pomts of AQ(K) hft to torsion points of

A defmed over an extension of K which is unramified at υ. In shorf for «-torsion

Abelian vaneties havmg purely additive reduction 295 We give some examples what happens if we consider points whose Order is di-visible by c h a r ( £ ) = / » 0 . Also in case of stable reduction it is not so difficult to describe the Situation (</[/?] ->Spec(/?) is quasi-finite in that case). Thus we suppose the reduction is purely additive; in that case all points on the connected component A0 of the special fibre A0 are />-power torsion, and s/[p]->Spec(/?) need not be quasi-finite. We use the filtration on E(K) äs introduced in [5, Section 4],

where

E(K)m = {(x, y) e E(K) \ υ(χ) < -2m, u(y) <

after having chosen a minimal equation for E.

4.1.1. Remark. We take/?>3. If PeE(K) (and ord(P)=,p = criar(£), and E has

ad-ditive reduction at D), then PeE(K)Q (because p>3 does not divide the number of connected components of E0, and Ε(Κ)0-+Ε®(£), use p. 46, table of [5]). We show that both cases ΡξΕ(Κ)ί and PeE(K){ indeed occur:

4.1.2. Example. Take p>3, we construct ΡεΕ(Κ), ord(P)=p and P$E(K){. Let

E be the curve 150. C (cf. [5, p. 103]), thus the curve given by the minimal equation

it has additive reduction at υ = υ5 (because 52 divides its conductor 150), and it has

a point of order 5 (indeed #E(Q) = 10). We claim

(relative the valuation ü5). This we can prove äs follows: by Remark 4.1.1 we know PeE(Q)0, thus the group (P) = NcE extends flatly to a finite group scheme /KOi over Spec(Z(5)) (one can work with the Neron minimal model o, but also

with the (plane) Weierstrass minimal model, and then Λ (χ) F5 is not the Singular

point because of PeE(<Q)0). If we woukl have PeE(Q)lt then it would follow

as= -^®F5 (because of additive reduction), but a5 over F5 does not lift to the

unramified Situation Z( 5 )-*F5 (cf. [18, Section 5]), thus

/>$£(<[}),.

One can avold the abstract proof by an explicit computation:

the tangent line at P is y = 20, so -2P=(8,20); the tangent line at -2P is

3X- 7 - 4 = 0, so 4P = (~4, -16) = - P , thus </>> = Z/5; the singular point on

^inodS is (x = 2, y = -\) mod5, thus PeE(<£))0, and the example is established.

296 H. W. Lenstra, Jr., F. Oort

4.1.1), thus p-PeE(K){ (because E has additive reduction), thus Q:=p'~lPe

E(K)} and ord(Q) =p.

Next we choose p = 3, and we show various possibilities indeed occur:

4.2.1. Example. We construct PeE(Q), with ord(P) = 3, P$E(<Q)0. Lei E be given

by the equation

Y2+3aXY+3bY=X3;

by well-known formulas (cf. [5, p. 36]) one computes

A = 36b\a3-3b).

If 36 does not divide b3(a3 — 3b), this equation is minimal (e.g. take a=l=b).

Fur-thermore P = (0, 0) is a flex on E (hence ord(P) = 3), and E mod 3 has a cusp al (0, 0). Thus P $ £ ( Q )0.

4.2.2. Example. It is very easy to give PeE(K) with ord(P) = 3, PeE(K)0 and P$E(K)t. E.g.

P = (0,2) on Y2 = X* + 4

(cf. 108. A in [5, p. 95]) has this property, because (x=— l, ^ = 0) mod 3 is the Singular point on E mod 3, thus P reduces to a point on E$ but not to the identity. Another example:

P = (0,0) on 72+ Υ = Λ"3

(cf. 27. A in [5, p. 83]) is a flex, which does not reduce to the cusp (x= l, y= 1) mod 3 on £"mod 3.

4.2.3. Example. We construct PeE(K) with ord(P) = 9, P$E(K)Q and 3P$E(K),.

Indeed consider K = <£), D = D3, and lake 54. B (cf. [5, p. 87]), a curve which has

ad-ditive reduction at 3 such that #£"(Q) = 9. Note that Q does not contain a primitive cube root of unity, thus £(Q) does not contain (Z/3) χ (Z/3), hence

let P be a generator for this group. Note that a3 over F3 does not lift to Z( 3 ), thus

P and 3P do not reduce to the identity under reduction modulo 3, hence

is injective, thus ord(P) = 9, and note lhat the extension

0 -NEW )0->£·((Ρ)->· Z/3

Abehan vaneties havmg purely additive leduction 297

4.2.4. Remark. Take ; = 3 m Example 3.7; then p = 3, PeE(K), ord(P) = 33

and E has additive reduction at D. Then 3PeE(K)0,

thus Q: = 9P has the property ord(Q) = 3,

4.3. Example. We conclude by an example with p = 2. Consider 48. E (cf . [5, p. 86]),

i.e

Y2=X3+X2+16X+\80;

the nght hand side factors over Q m the irreducible factors (X+5)(X2-4X+36),

hence £[2](Q) = Z/2 Because #£(Q) = 8 we conclude

(of course it is well-known that such examples exist, e g cf. [6, p. 35, Theorem 8]). Thus

£(Q),=0, £(<p)0=Z/2 = <Q = (5,0)>

and

(because (0, 0) mod 2 is the cusp on E mod 2, and Q mod 2 is smooth on E mod 2).

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[1] A Fröhlich, Local ficlds, in J W S Cassels and A Fröhlich, eds , Algebraic Number Theory (Academic Press, New York, 1967)

[2] A Grothendicck, M Raynaud and D S Rim, Seminaire de Geometrie Algebnque, SGA 7 I, 1967-1969, Le-ture Notes m Math 288 (Springer, Berlin, 1972)

[3] S Lang, Abdian Vaneties (Interscience, New York, 1959)

[4] Modular functions of onc vauable II (Antweip, 1972), Lecture Notes m Math 349 (Springer, Beilm, 1973) Especially P Dehgne and M Rapoport, Les Schemas de modules de courbes elhpti-ques, pp 143-316

[5] Modulai functions of one variable IV (Antweip, 1972), Lectuie Notes in Math 476 (Spimger, Berlin, 1975) Especially J Täte, Algonthm for deteimming the type of a Singular fibre in an ellip-tic pencil, pp 33-52, Table l, pp 81-113

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[10] ] Oort, Fmite group schemes, lotal moduh for abelian vaneties and hftmg problems, Compositio Math 23(1971)265-296 Also m AlgebraicGeometry, Oslo 1970 (Wolters-Noordhoff, Groningen, 1972)

[11] F Oort, Good and stable reduction of abelian vaneties, Manuscr Math 11 (1974) 171-197 [12] J P Serre, Corps Locaux, Act Sc Ind 1296 (Hermann, Paris, 1962)

[13] I P Serre, Cohomologie Galoisienne Lecture Notes m Math 5 (Springer, Berlin, 1964) [14] J P Serre and J Täte, Good reduction of dbehan vaneties, Ann ot Math 88(1968)492-517 [15] G Shimura and Υ Tamyama, Complex multiphcation of abelian vancties and its apphcations to

number theory, Math Soc Japan (1961)

[16] G Shimura, On the field of rationality for an abeJian vanety, Ndgoya Math J 45(1972)167-178

[17] J H Silverman, The Neron fiber of abelian vaneties with potential good reduction, Math Ann 264 (1983) 1-3